Extension of the Adiabatic Theorem

This paper investigates and provides numerical and analytical support for a proposed extension of the adiabatic theorem to nonadiabatic quantum quenches within the same phase, demonstrating that the overlap between the initial ground state and post-quench eigenstates is maximized for the new ground state in both the transverse field Ising and ANNNI models.

The Gist

The Big Idea: The "Comfort Zone" of Quantum Systems

Imagine you have a very shy, sensitive person (let's call them Quantum) living in a specific house. This house represents their current state of mind or "phase."

The Old Rule (The Adiabatic Theorem):
If you want to move Quantum to a new house, you must do it very, very slowly. If you walk them there at a snail's pace, they will stay calm and remain in their "ground state" (their most comfortable, lowest-energy position). They won't get confused or jump around; they will just smoothly transition into the new house's layout. This is the famous Adiabatic Theorem.

The New Question (The Quantum Quench):
But what if you don't walk them slowly? What if you grab them and teleport them instantly to a new house? This is called a Quantum Quench. It's a sudden, jarring change.

The authors of this paper asked a fascinating question:

"If we teleport Quantum to a new house that is in the same neighborhood (same 'phase'), will they still feel most comfortable in the new house's living room (the new ground state), or will they be so confused that they feel more at home in the attic, the basement, or the guest room (excited states)?"

The Hypothesis: "The Best Match"

The authors proposed a conjecture (a guess based on logic):
If the old house and the new house are in the same neighborhood, the "best match" for Quantum's soul is always the new house's living room.

In physics terms: If you change the rules of the game suddenly, but you stay within the same "phase" of matter, the system's original state will overlap most strongly with the new ground state, not with any of the excited, chaotic states.

The Experiments: Testing the Theory

To test this, the authors used two different "neighborhoods" (mathematical models) to see if the rule held up.

1. The Transverse Field Ising Model (TFIM): The Perfect Neighborhood

Think of this as a simple, orderly row of houses where everyone holds hands in a line.

• The Test: The authors did the math perfectly (analytically) for this model.
• The Result: It worked! They proved that no matter how you teleport Quantum within this neighborhood, they always feel most at home in the new living room. The "overlap" with the ground state is always the highest.
• The Catch: If you teleport them across the neighborhood border into a totally different type of town (a different phase, like from a magnet to a non-magnet), the rule breaks. They get confused and might prefer the attic.

2. The ANNNI Model: The Complicated Neighborhood

This is a more complex neighborhood. Imagine houses where neighbors argue not just with the person next door, but also with the person two doors down. This creates "frustration" (conflicting desires).

• The Test: This model is too messy for a perfect math proof, so the authors used two methods:
  1. Special Case: They looked at a specific, simplified line in this neighborhood where the math works out. Result: The rule held true here too.
  2. Computer Simulation: They used supercomputers to simulate smaller versions of this neighborhood (since real quantum systems are infinite, they had to use small models).
• The Result: Mostly, the rule held true! However, in some specific spots near the "border" between neighborhoods, the computer simulations showed some confusion.
• The Explanation: The authors suspect these "violations" happen because their computer models were too small (like trying to judge a whole city by looking at a single block). In the real, infinite world, these violations likely disappear, and the rule would hold.

The Analogy of the "Soul" and the "Map"

Imagine the Ground State is a perfect, comfortable bed.

• Adiabatic (Slow change): You slowly move the bed to a new room. You stay in the bed.
• Quench (Sudden change): You suddenly drop the bed into a new room.
  • The Conjecture: Even though the drop was sudden, if the new room is similar to the old one, you will still land in the bed (the ground state). You won't accidentally land on the floor (an excited state).
  • The Finding: The paper says, "Yes, generally, you land in the bed, as long as the new room isn't a totally different type of building."

Why Does This Matter?

This isn't just about abstract math. It helps us understand:

1. Quantum Computers: When we program quantum computers, we often change their settings quickly. Knowing that the system stays "grounded" (in its lowest energy state) helps us predict if the computer will work or crash.
2. New Materials: It helps scientists understand how materials behave when they are shocked or changed rapidly, which is crucial for developing new technologies.

The Conclusion

The authors conclude that nature is surprisingly resilient. Even when you shake things up violently (a quench), if you don't change the fundamental nature of the system (the phase), the system tends to "remember" its calm, lowest-energy state.

They call this an "Extension of the Adiabatic Theorem." It's like saying: "The old rule said 'If you move slowly, you stay calm.' We found a new rule: 'Even if you move fast, as long as you stay in the same neighborhood, you'll still find your way back to the living room.'"

One small warning: This rule works best in the "real world" (infinite systems). In small, artificial simulations, it sometimes glitches near the borders of different phases, but the authors believe this is just a limitation of the simulation size, not a failure of the law of physics.

Technical Summary

1. Problem Statement

The paper investigates the validity of extending the quantum adiabatic theorem to the regime of quantum quenches (maximally non-adiabatic changes).

• Context: The standard adiabatic theorem (Born and Fock, 1928) states that if a system evolves slowly enough ($\tau \gg (\delta E)^{-1}$), it remains in its instantaneous eigenstate.
• The Conjecture: The authors propose a conjecture for non-adiabatic quenches: If a system is quenched from an initial Hamiltonian $H_i$ to a final Hamiltonian $H_f$ within the same phase (connected by a continuous path without gap closing), the overlap between the initial ground state $|GS_i\rangle$ and the post-quench eigenstates $|\psi_f^n\rangle$ is maximized when $|\psi_f^n\rangle$ is the post-quench ground state $|GS_f\rangle$.
  $$\max_n |\langle GS_i | \psi_f^n \rangle|^2 = |\langle GS_i | GS_f \rangle|^2$$
• Challenge: While intuitively plausible, this statement is not trivially true for non-adiabatic processes, especially in many-body systems near critical points or in complex phases. The paper seeks to prove or disprove this conjecture analytically and numerically.

2. Methodology

The authors employ a combination of analytical proofs and numerical simulations on two specific 1D quantum spin models:

• Models Studied:

  1. Transverse Field Ising Model (TFIM): An exactly solvable model.
  2. Axial Next Nearest Neighbor Ising (ANNNI) Model: A non-integrable model featuring frustration, next-nearest-neighbor interactions, and a complex phase diagram (Paramagnetic, Ferromagnetic, Floating, and Antiphase).

• Analytical Approach:

  • TFIM: Utilizes Jordan-Wigner and Bogoliubov transformations to map the spin system to free fermions. The authors derive the exact form of the ground state overlap and excited state overlaps in momentum space to test the conjecture analytically across the entire phase diagram.
  • ANNNI: Focuses on the Peschel-Emery (PE) line, a special disorder line where the Hamiltonian factorizes into local terms, allowing for an exact analytical solution of the ground state and specific excited states as product states.

• Numerical Approach:

  • Exact Diagonalization (ED): Used to compute the full spectrum and overlaps for the ANNNI model.
  • System Sizes: Limited to $L=16$ spins due to computational constraints.
  • Finite-Size Scaling: The authors analyze "finite-size shifted phase boundaries" using fidelity susceptibility ($\chi_F$) to determine if observed violations of the conjecture are artifacts of small system sizes or genuine physical phenomena.
  • Visualization: Overlaps are plotted against the post-quench energy spectrum to verify if the maximum overlap aligns with the ground state energy ($E=0$).

3. Key Contributions and Results

A. Transverse Field Ising Model (TFIM)

• Result: The conjecture is analytically proven to hold for all quenches within the same phase (both Paramagnetic and Ferromagnetic).
• Mechanism: The overlap with excited states involves factors of $\tan^2[\theta_k(h_i) - \theta_k(h_f)]$. The authors demonstrate that for any two parameters $h_i, h_f$ within the same phase, the difference in the Bogoliubov angles $|\theta_k(h_i) - \theta_k(h_f)|$ is strictly less than $\pi/4$.
• Implication: Since $\tan^2(x) < 1$ for $|x| < \pi/4$, the overlap with any excited state is strictly smaller than the overlap with the ground state. The conjecture fails only when crossing the critical point ($h=1$).

B. ANNNI Model (Special Case)

• Result: The conjecture is analytically proven for quenches occurring strictly along the Peschel-Emery (PE) line.
• Mechanism: Along this line, the ground state is a product state. The ratio of the overlap with an excited state to the overlap with the ground state is shown to be less than 1 for all relevant parameter limits ($\alpha \in [1, \infty)$).

C. ANNNI Model (General Case - Numerical)

• General Findings: Numerical simulations generally support the conjecture for quenches deep within the Paramagnetic, Ferromagnetic, Floating, and Antiphase regions.
• Violations:
  • Violations occur when the quench starts near phase boundaries (specifically near the quadruple point or Ising transition) in the Paramagnetic phase.
  • In these regions, some excited states exhibit a larger overlap with the initial ground state than the final ground state does.
• Finite-Size Effects: The authors investigate whether these violations are due to the small system size ($L=16$).
  • They calculate finite-size shifted phase boundaries using fidelity susceptibility.
  • Conclusion: Some violations are explained by the shift in phase boundaries (the quench effectively crosses a "finite-size" phase transition). However, some violations remain unexplained even after accounting for these shifts, suggesting the conjecture may not hold universally near critical points or in complex phases like the Floating phase.

D. Spectral Analysis

• When plotting overlaps against the post-quench energy spectrum, quenches within the same phase typically show the highest peak at the ground state energy.
• Quenches crossing phase transitions (e.g., Paramagnetic to Ferromagnetic) consistently show the maximum overlap at higher energy levels, confirming that the conjecture is specific to "same-phase" quenches.

4. Significance and Conclusion

• Extension of Adiabaticity: The work suggests that the "memory" of the initial ground state is preserved most strongly in the final ground state, even under sudden (non-adiabatic) changes, provided the system remains within the same thermodynamic phase. This extends the conceptual reach of the adiabatic theorem to maximally non-adiabatic regimes.
• Role of Criticality: The study highlights that the conjecture breaks down near quantum critical points and phase boundaries, where the fidelity susceptibility diverges and the ground state changes rapidly.
• Limitations: The authors acknowledge that while the conjecture holds for the TFIM and specific ANNNI cases, it is not rigorously proven for all systems. Counterexamples exist in finite systems or near complex phase boundaries (like the floating phase in ANNNI).
• Future Outlook: The paper concludes that the conjecture likely holds for a "significant class" of Hamiltonians in the thermodynamic limit, far from critical points. However, determining the precise boundaries of its validity (e.g., distance from critical points, specific symmetry classes) remains an open problem requiring further investigation.

In summary, Damerow and Kehrein provide strong evidence that ground state fidelity is maximized for the ground state during same-phase quenches, validating a non-adiabatic extension of the adiabatic theorem, while identifying critical regions where this intuition fails.